Q-learning, am I interpreting correctly $Q(s,a) = r + \gamma \max_{a'} Q(s',a')$?
What normally happens in DQN is the following:

First, the NN is used to estimate an approximation of Q for each state-action pair by using a vector of weights:
$$Q(s,a,w) \simeq Q(s,a)$$

There are two options: You can either feed it a certain state, and get as output the value of each possible action, or, you can input a state-action pair and get as output the value of that pair. (basically the NN gives you the policy)

After having this output you can now use your environment to simulate the next state. So basically you apply the action given by the NN (maybe using $\epsilon -greedy$), and you get the next state, and your next reward from your environment by observing it.

Where does the Q function comes in all of this? The Q-function 
$$Q(s,a) = Q(s,a) + \alpha [r(s,a) + \gamma  max_a{Q(s',a)} - Q(s,a)]$$
that in standard Q-learning is used to update the values after each simulation step, is now used as the Target of the neural network, to make the back propagation.

There are some considerations you have to make to help this converge, mainly experience replay and fixed Q-targets.

So when you say " I don't have yet the next state", is because you need to simulate the action that is selected to get this next state.

EDIT: The main thing you might be missing is that the right part of the equation you present in the title is called TD-Target, and it is not used to make the updates, you might see it as the final result you want to achieve $Q^*(s,a)$ (the optimal values, after convergence). But to make the update you need to use the weighted function with the previous value of $Q(s,a)$ as I showed above, using a learning rate $0&lt; \alpha &lt; 1$